It is not hard to check that \((T_{\beta }, \mu _{\beta }, \mathcal {A}_{\beta })\)  is a RU map. Also, it is known that \(T_{\beta }\)  is strongly mixing (see Section 4.5 in [1]} or Theorem 1C in [2]}), hence weakly mixing, so \((T_{\beta }, \mu _{\beta }, \mathcal {A}_{\beta })\)  has property M by Theorem REF .
